(will be inserted by the editor)
Information Variability Impacts in Auctions
Justin Jia · Ronald M. Harstad · Michael H. Rothkopf
the date of receipt and acceptance should be inserted later
Abstract A wide variety of auction models exhibit close relationships be-tween the winner’s expected profit and the expected difference bebe-tween the highest and second-highest order statistics of bidders’ information, and be-tween expected revenue and the second-highest order statistic of bidders’ ex-pected asset values. We use stochastic orderings to see when greater envi-ronmental variability of bidders’ information enhances expected profit and expected revenue.
Keywords auctions, revenue, expected profit, order statistics, stochastic orderings (JEL:D44)
In the usual formulation of a game of incomplete information, each player privately observes his type, with each type drawn from a distribution that is common knowledge. Auction models typically follow this specification, either with types drawn independently, or drawn from affiliated distributions; inde-pendence conditional on a stochastic common value is a frequent example. A bidder values the auctioned asset as some commonly known function of the profile of types, with common-value auctions and private-values auctions (in which each bidder’s value is his type) as special cases.
Typically, the equilibrium examined in these models exhibits a positive expected profit conditional on winning.1 The winner’s profit results from the presence of private information: his estimate of asset value, properly condi-tioned on his type and on the event of winning, is an estimate for which other
Justin Jia
Smeal College of Business, Pennsylvania State University. E-mail: [email protected] Ronald M. Harstad
Department of Economics, University of Missouri. E-mail: [email protected] Michael H. Rothkopf
Smeal College of Business, Pennsylvania State University, deceased February 2008.
1 The exceptions stem from various heroic assumptions yielding full surplus extraction,
bidders have but an imperfect substitute. The most widely studied case is inde-pendently drawn types; then under mild assumptions, Myerson (1981) shows that equilibrium expected revenue for any standard auction form is equal to the second-highest expected value estimate, and expected profit conditional on winning equals the difference between the highest and second-highest expected value estimates. This paper examines the relationship between the variability of the type distributions and these expected estimates, illuminating compar-isons across auction environments where bidders’ private information is less dispersed in one, and dispersed more widely in the other.2
To normalize to zero the seller’s value of the auctioned asset is usual, harmless, and assumed throughout. The gains from trade are then equal to the highest expected value estimate. Except in virtually pathological cases, more widely dispersed private information increases the gains from trade. We explore assumptions determining whether the seller, and whether the winning bidder, shares in these increased gains.
In some auction settings, variability of bidders’ information may be par-tially under the bid-taker’s control: a seller may decide to reveal greater or lesser information about technical specifications of an auctioned asset; a buyer seeking bids for construction of a skyscraper may choose to offer insurance against price increases for certain raw materials as part of the auction speci-fications.
Sections 1 and 2 examine independently-distributed information; affiliation is considered following. Most proofs are relegated to the appendix.
1 The Winning Bidder’s Expected Profit
This analysis contrasts a benchmark environment where any of thenbidders observes a type X drawn i.i.d. from nondegenerate distribution function F
with a more-dispersed-information environment where any bidder observes a typeZ =X+Y, with the added randomness Y distributed (independently) as G. 3 Let X = X
(1), ..., X(n)
denote the order statistics (in descending order) of a sample of size n from the population random variable X, and Z= Z(1), ..., Z(n)
, correspondingly. The characterization sought,Dispersion Benefits Bidders (DBB), is then formally:
EX(1)−X(2)
≤EZ(1)−Z(2)
.
Several types of stochastic orderings play roles:
2 For concreteness, the text develops the case where bidders compete to buy from a
bid-taking seller; completely corresponding results about the variability of (1) the second-lowest order statistic, and (2) the difference between the lowest and second-lowest order statistics, illuminate expected project cost and expected profit in the case where bidders compete to sell to a bid-taking buyer.
3 The incremental dispersion need not be additive:Zcan simply be more dispersed than
1. X isstochastically smaller thanY (read, X≤stY) ifF(t)≤G(t)∀t.4 2. X isless dispersed than Y (X ≤disp Y) if 0 < α < β <1 =⇒F−1(β)−
F−1(α)≤G−1(β)−G−1(α) (Shaked 1982).
3. X is smaller thanY in theexcess wealth order (X≤ewY) ifp∈(0,1) =⇒
R∞
F−1(p)F(x)dx≤
R∞
G−1(p)G(x)dx (Shaked and Shanthikumar 2006). 4. Xis smaller thanY in theincreasing convex order(X ≤icxY) if
R∞
t F(x)dx≤
R∞
t G(x)dx ∀t(Ross 1996).
Theorem 1 Either (1)X has a log-concave density5or (2)X has an increas-ing failure rate (IFR)6 is sufficient for DBB.
Proof Preliminaries: [a] X satisfies X ≤disp X +Y for Y independent of X if and only if X has a log-concave density. [b] Let V(i) = X(i)−X(i+1) and
W(i) = Z(i)−Z(i+1) , i = 1,2, ..., n.7 If X ≤disp Z, then V(i) ≤st W(i),
i= 1,2, ..., n. [c]X satisfiesX ≤ewX+Y forY independent ofX if and only ifXis IFR. [d] IfX ≤ewZthenX(1)−X(k)≤icxZ(1)−Z(k)fork= 2,3, ..., n.8 If X has a log-concave density, then (by [a]) X ≤disp X +Y, and (by [b]) V(1) ≤st W(1). Now E V(1) ≤ EW(1) follows from {A≤stB} =⇒
{E[A]≤E[B]}. If X is IFR, then (by [c]) X ≤ew X +Y, and (by [d])
X(1)−X(2) ≤icx Z(1)−Z(2), which is a stronger result thanE
X(1)−X(2) ≤ E Z(1)−Z(2) .9
Many commonly used distribution families have log-concave densities. These include uniform, normal, logistic, extreme value, exponential, Laplace, chi-squared (c ≥2), Weibull (c ≥1), power function (β ≥ 1) , gamma (m ≥1) and beta (a≥1, b≥1) (See Bagnoli and Bergstrom 2005 for a more extensive list).
It is frequently the case in auction theory that results for two-bidder auc-tions may only extend to aucauc-tions wheren >2 under more stringent conditions (if at all). This is such a situation, as the above conditions are unnecessary in two-bidder auctions:
Theorem 2 Thatn= 2 is sufficient for DBB.
4 For a random variableW∼H, the survival functionH(t) = Pr{W > t} ∀t.
5 Paul and Gutierrez (2004) obtain a corresponding characterization under a more
strin-gent and less transparent assumption on the underlying stochastic order; their principal concern is with the relationship between the number of bidders and the expectation of the difference between the two highest order statistics.
6 Log-concave density and IFR are related but not nested. If the density function h is
continuously differentiable and log-concave on [a, b], then the distributionH has IFR.
7 For completeness, for U ∼H, defineU
(n+1) = inf{t :U(t) >0}; this step assumes
X(n+1)andZ(n+1) are finite, which is harmless for present purposes.
8 Sources: [a] Shaked and Shanthikumar 2006 Theorem 3.B.7; [b] Bartoszewicz 1986
Lemma 3(b); [c] Shaked and Shanthikumar 2006 Theorem 3.C.8; [d] Kochar et al. 2007 Theorem 3.
These three conditions suffice for dispersion to benefit bidders, but do not indicate the extent of the benefit. A fourth condition offers a measure of bidder benefit.
Theorem 3 ThatX and Y are normal is sufficient for DBB. Moreover, the increment of expected profit is proportional to the increment of the standard deviation.
None of these sufficient conditions are necessary. An abundance of cases exhibit DBB for arbitrary n, but no illuminating necessary condition seems possible.
It might appear that the expected difference between the two highest order statistics should always expand as variability increases. A counterexample: let
X ∼log-normal(2,4),Y ∼N(15,5); a 10,000-iteration simulation withn= 50 yieldsE X(1)−X(2) ≈8.49, butE Z(1)−Z(2) ≈7.57.
To understand how the example works, we introduce the concept oflocally dispersive order. We say thatXislocally less dispersed thanY on range subset [α, β] if 0< α < β <1 and ∃d∈(−α,1−β) such that F−1(β)−F−1(α)≤
G−1(β+d)−G−1(α+d). Intuitively, the definition means that the C.D.F. of
Y is flatter than that ofX on some regions on their respective supports. In our problem, althoughZ tends to be more dispersive thanXglobally, the converse can happen locally. This leads to an algorithm to construct counterexamples: first, select Y such that Z is locally less dispersed than X; then select n
such that X(1),X(2), and Z(1), Z(2) fall in the corresponding regions. In the above counterexample,E X(1) ≈18.90 andE X(2)
≈10.41, both far in the tail of the distribution. Adding a normal distribution with a mean between
EX(1)
and EX(2)
increases the variability, but also shifts rightward the log-normal distribution’s head. The first- and second-order statistics are then in the resultant distribution’s head. As the tail is much flatter than the head, the desired locally dispersive order is obtained. The general structure of cases reversing the DBB inequality eludes specification. The example demonstrates that dispersion does not always benefit the winning bidder.
2 Expected Revenue
As stated in the introduction, the bid-taker’s expected revenue is the second-highest expected value estimate. Continuing withZ =X+Y, X, Y indepen-dent, we explore conditions yielding the characterizationDispersion Benefits Seller (DBS), formally: E X(2) ≤E Z(2) .
First, some intuition. Unlike E
W(1)−W(2)
, the expected value of the second-highest order statistic is clearly location dependent. For example, if
X ∼U[0,1] andY ∼U[−3,−2], then obviouslyZ(2) ≤ −1. Roughly speaking, the addition of Y can be thought to have two effects. First, the mean of Y: ifE[Y]>0, it can “push up”EZ(2)
may “drag it down.” Second, the variability ofY: it increases the variability of
Z. This will approximately “push up” all order statistics above the mean and “push down” those below it. For n >3, greater variability tends to increase the second order statistic–usually above the mean.
Proposition 1 ThatY is nonnegative is sufficient for DBS.
It is trivial to construct a case in which DBS fails by choosing negative
Y. To avoid trivialities, restrict attention to Y with E[Y]≥0. Whenn= 2, the second order statistic is below the mean as long as the distribution is not very right-skewed. To find a counter case is easy. Whenn= 3, EZ(2)
is the median. To construct a similar example, focus on the right-skewed distribu-tions, as the mean exceeds the median. Let X ∼log-normal(10,10), T ∼ log-normal(10,100) and Y =T−9, so E[Y] = 1. By 10,000-iteration simulation withn= 3, EX(2)
≈8.25< EZ(2)
≈3.69. This inequality holds as well in simulations forn= 4,5,6.
So examples of higher revenue with less dispersed information exist when
n≤6. How about many bidders? Intuitively, the largernis, the more difficult to find such an example. While the natural conjecture is that examples may be constructed correspondingly, we know of no algorithm assuredly constructing examples for arbitrarily largen.10
3 Going Beyond Independent Randomnesses
This section moves from independent randomness to the more complicated environments in which randomnesses are correlated. We consider a widely used case of the affiliation model: conditionally independent random variables. Let there be a random variable V, and let (X1, . . . , Xn) and (Y1, . . . , Yn) be independent conditional onV =v(Y being white noise is included as a special case).
The previous two sections utilized tools that link stochastic orders to or-der statistics unor-der that assumption of independence; to our knowledge, there are not extant tools for analyzing affiliated variables. Accordingly, the char-acterizations here are slightly narrower than for independent types (although Proposition 1 extends by the same proof).
In general, if sufficient conditions in section 1 [2] above hold givenV =v∀v
for conditional variablesX andY, then the same conditions are uncondition-ally sufficient for DBB [DBS] with affiliated values. These extensions follow from the relationshipEA[A] =EB[E[A|B]]. To characterize fully general con-ditions under which sufficient concon-ditions hold “for each v” is elusive. Two broad, fairly general classes are treated in the following two theorems:
Theorem 4 If X has an unconditional distribution that is hierarchical, that is, ifV is a parameter of the distribution ofX, then sufficient conditions for
DBB [DBS] under independent types remain sufficient when X and Y are independent conditional onV.
Letting V be the asymptotic mean of X is sufficient to satisfy Theorem 4 when (X1, . . . , Xn) are exchangeable. For example, if X ∼ normal(V, σ), then X has log-concave density for all V = v, extending DBB. Hierarchical distribution is sufficient but not necessary:
Theorem 5 If X = W +g(V) for some nondecreasing function g(·), then either a log-concave density or IFR ofWi’s is sufficient for DBB.
Proof X +Y = W +g(V) +Y. The claim follows immediately by treating
g(V) +Y as a random variable and applying Theorem 1.
A situation fitting Theorem 5 is when the private-value and common-value components are additive (as in, e.g., Pesendorfer and Swinkels 2000, or Goeree and Offerman 2003).11
Section 1 [2] provided examples in which dispersion harmed bidders [seller] whenX andY are independent. Corresponding examples for affiliatedX0and
Y0 can easily be constructed in the following way: take any of the examples favoring the less dispersed environment for independent X and Y. Letting
X0 = X and Y0 = Y +εX0 introduces affiliation. Letting |ε| be arbitrarily small maintains affiliation and the inequality found above for that example.
4 Concluding Remarks
Under tenable assumptions for independent-types or affiliated-types environ-ments, greater dispersion in bidders’ information aids both bidders, via higher expected profit, and sellers, via higher expected revenue. Yet the needed as-sumptions are far from innocuous, as unusual but not pathological cases have been presented where greater variability in information can yield the reverse inequalities for expected differences between the two highest order statistics and for expectations of the second-highest value estimate.
There is reason to expect some robustness of the conclusions drawn, in such directions as small deviations from symmetric type distributions or from distributional families discussed. An example with sizable deviations from symmetric type distributions illustrates. Let types be drawn from uniform distributions: X1 ∼ U[2,6], X2 ∼ U[3,6], X3 ∼ U[4,6], X4 ∼ U[5,6] and the mean-preserving spreads Z1 ∼U[1,7], Z2 ∼U[2,7], Z3 ∼U[3,7], Z4 ∼
U[4,7]. The more dispersed environment exhibits a greater expected profit,
E X(1)−X(2) = 0.45< E Z(1)−Z(2)
= 0.83, and a greater expected rev-enue,EX(2)
= 5.21< EZ(2)
= 5.34. The greater expected profit in the higher variability environment wouldn’t necessarily be an expectation each bidder could face, but in this example, conditional on winning, each bidder’s expected profit is higher with type distributions (Zj) than with (Xj).12
11 The appendix considers situations where affiliated types do not satisfy conditional
in-dependence.
As noted, often bid-takers may have auction design decisions and information-disclosure policy decisions that can impact type variability. There may also be scope for decisions by bidders that have information-dispersion aspects. Matthews (1977, 1984) and Bergemann and Valimaki (2002) consider mod-els in which bidders decide how precise an estimate of asset value to acquire, at a cost; more precise estimates are likely to yield the environments con-sidered here to have less dispersed types. In their models, ex-ante symmet-ric information-improving technologies lead to equilibria in which information precision is symmetric. The results above suggest the circumstances in which greater information-improving efforts might coincide with lesser (or greater) dispersion of resulting asset value estimates and thus with lesser or greater gra-dients of expected profitability. Further, they suggest when a seller might have an incentive to subsidize (or, if feasible, to tax) bidders’ information-improving costs.
Appendix
Proof of Theorem 2:Construct trial observations ofZas follows. Take 2 draws ofX, denoted
x(1), x(2) (descendingly), and 2 draws of Y, denoted byy(1), y(2). Next, randomly assign
y(i)’s tox(i)’s, i= 1,2. It is equally likely that x(1) is associated withy(1) or y(2), each
having probability 0.5. First, supposex(1)is associated withy(1), sox(2)is associated with
y(2). Recall z(1) = x(1)+y(1) and z(2) = x(2)+y(2), so ∆z = z(1)−z(2) = (x(1)−
x(2)) + (y(1)−y(2)) =∆x+∆y. Second, if insteadx(1)is associated withy(2), thenx(2)
is associated withy(1). It is then unclear which of x(1)+y(2) and x(2)+y(1) is bigger,
but clear that∆z=|(x(1)+y(2))−(x(2)+y(1))|=|∆x−∆y|. Hence the expected value
of ∆z = 0.5(∆x+∆y) + 0.5|∆x−∆y| ≥ 0.5(∆x+∆y) + 0.5(∆x−∆y) = ∆x, namely
∆z ≥ ∆xin each such constructed trial. Then, by the law of large numbers, E[∆Z] = limn→∞ Pn k=1E[∆zk] n ≥limn→∞ Pn k=1∆xk n =E[∆X].
Proof of Theorem 3:LettingZ=αX+β, whereαis a positive constant, it is straight-forward that
1. Z(i),αX(i)+βfor alli, and13
2. Z(1)−Z(2),α
X(1)−X(2)
.
Ifα= 1 andβ=−µ=−E[X], thenZ(1)−Z(2)=X(1)−X(2)forZ=X−µ. Thus the
difference between the first- and second-order statistics is location independent. Normalizing the means of the random variables to 0 is without loss of generality.
To show Theorem 3, assume both X and Y have means 0. By closure of normal distributions under addition, Z = X+Y is also normally distributed with mean 0. Let
α=qσ2
x+σ2y/σx, thenαXis a normal random variable having the same mean and
vari-ance asZ. Hence,Z,αX, soE Z(1)−Z(2) =αE X(1)−X(2) .
Proof of Proposition 1: Letx1, ..., xn be a size-nrealization ofX, and x(1), x(2) the
two largest values. Adding a size-nrealization ofY tox1, ..., xn yieldsz1, ..., zn. Suppose
x(1), x(2)becomex(1)+ξ, x(2)+τ, whereξ, τ ≥0 by assumption. Clearly,
z(1), z(2) are
not necessarily
x(1)+ξ, x(2)+τ . Ifz(i)=x(2)+τfor somei= 2, ..., n, thenz(2)≥z(i)=
x(2)+τ ≥x(2). Ifz(1) =x(2)+τ, thenz(2) ≥x(1)+ξ≥x(2). Since this is true in each
realization,E X(2) ≤E Z(2)
. Note that the proof does not requireY to be independent ofX.
Further notes on Section 2:An example: IfX∼N(µx, σx2) ,Y ∼N(µy, σy2), a sufficient
condition for DBS is that (1−qσ2
x+σy2/σx)µx+µy>0. To see this, note that by closure
under addition,Z=X+Y is also a normal. Letα=qσ2
x+σ2y/σxandβ= (1−α)µx+µy,
thenZ,αX+β. Note thatα≥1. Ifβ >0, thenE
Z(2) =αE X(2) +β > E X(2) . Several questions: (1) Does DBS follow ifX is symmetric,Y has mean 0 , andn >3? No, an example:X ∼ [1 +N(0,1)] andY ∼ log−normal(2,4)−2, E
Z(2) ≈0.98< E X(2)
≈1.30. This is due to the skewness ofY. (2) Does DBS follow if in additionY is symmetric? While our conjectured answer is no, no verifying example is at hand. However, ifX has a log-concave density, then DBS follows, since the space between any two adjacent order statistics increases (this is preliminary [b] in the proof of Theorem 1).
Further notes on Section 3: That the joint distribution (X, Y) is bivariate normal is sufficient for DBB. If, in addition, (1−qσ2
x+σy2+ 2ρxyσxσy/σx)µx+µy > 0, (where
µx, µy, σx, σy, ρxy are the means, standard deviations, and correlation coefficient), DBS
follows. To see this, note that if the joint distribution (X, Y) is bivariate normal, then
Z=X+Y is normal. Lettingα=qσ2
x+σy2+ 2ρxyσxσy/σxandβ= (1−α)µx+µyyields
Z,αX+β. DBB follows sinceα≥1. In addition, ifβ >0, thenE
Z(2) > E X(2) . An alternative construction is specialized, but not distribution-specific. Let X and Y
be independent, and assume that forZ=X+Y, sufficient conditions for DBB and DBS apply (from any results in sections 1 and 2). ConstructX0=Xand
Y0=
αX0+β, with probabilityp Y, with probability 1−p ,
whereα, β≥0, p∈(0,1). By construction,X0andY0are affiliated. Then,
Z0=X0+Y0=
(α+ 1)X+β, with probabilityp X+Y, with probability 1−p .
andα0=α+ 1≥1, β≥0 suffice for DBB and DBS in this affiliation setting.
Another example of asymmetric distributions: Types are draws from normal distribu-tions: X1 ∼ N(10,4.5), X2 ∼ N(10.5,4), X3 ∼ N(11,3.5), X4 ∼ N(11.5,3), and the
variability that the less dispersed environment adds is symmetric: Zj = Xj+Y, with
Y ∼N(0,5), again a mean-preserving spread. This example reaches the same comparisons:
E X(1)−X(2) = 2.71 < E Z(1)−Z(2) = 4.53, andE X(2) = 11.93 < E Z(2) = 12.67. References
1. Bagnoli, Mark and Ted Bergstrom, 2005, Log-concave probability and its applications,
Economic Theory, 26, pp. 445-469.
2. Bartoszewicz, Jaroslaw, 1986, Dispersive ordering and the total time on test transfor-mation,Statistics & Probability Letters, 4, pp. 285-288.
3. Bergemann, Dirk, and Juuso Valimaki, 2002, Information acquisition and efficient auc-tion design,Econometrica, 70(3), pp. 1007-1033.
4. Goeree, Jacob, and Theo Offerman, 2003, Competitive bidding in auctions with private and common values,Economic Journal, 113, pp. 1-20.
5. Kochar, Subhash, Xiaohu Li, and Maochao Xu, 2007, Excess wealth order and sample spacings,Statistical Methodology, 4, pp. 385–392.
6. Li, Xiaohu, 2005, A note on expected rent in auction theory,Operations Research Letters
33, pp. 531 – 534.
7. Mares, Vlad, and Ronald M. Harstad, 2007, Ex-post full surplus extraction, straight-forwardly,Economic Theory, 32, pp. 399-410.
8. Matthews, Steven, 1977, Information acquisition in competitive bidding processes, work-ing paper, California Institute of Technology.
9. Matthews, Steven, 1984, Information acquisition in discriminatory auctions,Bayesian Models in Economic Theory, M. Boyer and R. E. Kihlstrom, eds. North-Holland, pp. 181-207.
10. Myerson, Roger B., 1981, Optimal auction design,Math of Oper. Res., 6, pp. 58-73. 11. Paul, Anand, and Genaro Gutierrez, 2004, Mean sample spacings, sample size and
variability in an auction-theoretic framework, Operations Research Letters, 32, pp. 103-108.
12. Pesendorfer, Wolfgang, and Jeroen Swinkels, 2000, Efficiency and information aggrega-tion in aucaggrega-tions,American Economic Review, 90, pp. 499-525.
13. Ross, S. M., 1996,Stochastic Processes (2nd ed.), Wiley, New York.
14. Shaked, M., 1982, Dispersive Ordering of Distributions,Journal of Applied Probability, Vol. 19, No. 2, pp. 310-320
15. Shaked, M. and J. G. Shanthikumar, 2006, Stochastic Orders, Springer Verlag, New York.
